To determine the accumulated value (A) of an investment using the compound interest formula, we need to use the formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount (initial investment),
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form),
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year,
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
Given:
- [tex]\( P = \$10,000 \)[/tex]
- [tex]\( r = 6.5\% = 0.065 \)[/tex]
- [tex]\( n \)[/tex] = 2 (since the interest is compounded semiannually)
- [tex]\( t = 3 \)[/tex] years
Now we plug these values into the formula:
[tex]\[ A = 10000 \left(1 + \frac{0.065}{2}\right)^{2 \cdot 3} \][/tex]
First, calculate the interest rate per compounding period:
[tex]\[ \frac{0.065}{2} = 0.0325 \][/tex]
Then add 1 to this value:
[tex]\[ 1 + 0.0325 = 1.0325 \][/tex]
Raise this to the power of the total number of compounding periods:
[tex]\[ 1.0325^{6} \][/tex]
Now, multiplying by the principal amount:
[tex]\[ A = 10000 \times (1.0325^6) \][/tex]
Through detailed calculations, the accumulated value (rounded to the nearest cent) after 3 years with semiannual compounding is:
[tex]\[ \boxed{12115.47} \][/tex]
So, the accumulated value, if the money is compounded semiannually, is $12,115.47.
